Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = \frac { 1 } { 4 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = - \frac { 1 } { 2 }$, with $\mathrm { t }$ in quadrant IV.
A) $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$
B) $\frac { 1 + 3 \sqrt { 5 } } { 8 }$
C) $\frac { 1 - 3 \sqrt { 5 } } { 8 }$
D) $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$

Question

Quizplus · Accepted Answer

To find $\tan(s + t)$, we use the identity $\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}$. First, we find $\tan s$ and $\tan t$. Since $\sin s = \frac{1}{4}$ and $s$ is in quadrant II, where sine is positive and cosine is negative, we use the Pythagorean identity $\sin^2 s + \cos^2 s = 1$ to find $\cos s$, and then $\tan s = \frac{\sin s}{\cos s}$. Similarly, for $t$ in quadrant IV, where sine is negative and cosine is positive, we find $\cos t$ and then $\tan t$. After calculating $\tan s$ and $\tan t$, we substitute these values into the $\tan(s + t)$ formula to get the exact value.

Find the Exact Value of the Expression Using the Provided tan⁡(s+t)\tan ( \mathrm { s } + \mathrm { t } )tan(s+t)

Find the Exact Value of the Expression Using the Provided $\tan ( \mathrm { s } + \mathrm { t } )$